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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
NEW ADVANCES IN SHELL MODEL
CALCULATIONS: APPLICATIONS AROUND
DOUBLY MAGIC NUCLEI40Ca AND
132Sn
Houda NAÏDJA
IPHC Strasbourg, FranceGSI Darmstadt, Germany.
LPMS, Université Constantine 1, Algeria.
ICNT Program: Theory for open-shell near the limits of stability,
May 11-29, 2015
H. Naïdja IPHC
ICNT
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Outline
PART 1 : AROUND40Ca
1. Ca isotopes shift
2. Isomer shift in 38K
3. Effect of pairing correlations
PART 2 : AROUND132Sn
1. Choose Valence space/Core
2. develop an effective interaction
3. Calculation of the energies, B(E2)
and masses in 134,136,138Sn,◮ Effect of core excitations◮ Closure or no of the sub-shell at
N=90
4. Other applications to the open n-p
systems : Te, Xe, Ba, Ce and Nd.
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
PART I : ISOMER SHIFT IN38K
3/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Ca isotopes shift
REMINDER ON Ca ISOTOPES SHIFT
νP
O16
d5/2 d5/2
s1/2
d3/2
p3/2
f7/2
s1/2
d3/2
f7/2
p3/2���
���
���
���
���
���
���
���
4/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Ca isotopes shift
REMINDER ON Ca ISOTOPES SHIFT
ZBM2 interaction :◮ based on realistic TBME
◮ monopole corrections to ensure 40Ca and48Ca gaps
◮ full space calculations
◮ almost free of center of masscontamination
◮ provides very good spectroscopy at sd-pfinterface
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Ca isotopes shift
✞
✝
☎
✆δ r2
c = 1Z nπ
fp(A)b2
{
nπfp
π occupation probability in fpshell
b = 1.974fm oscillateur parameter
39 40 41 42 43 44 45 46 47 48 49A
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
r p
2 (A)−
r p
2 (40)
(fm
2 )
Isotope shifts in Ca chain
exp sm
Phys.Lett B 522, 240 (2001)
The parabolic dependence on A is
related to the partial breaking of the
Z = 20 shell closure due to the πνcorrelations
correlation energies
40Ca 42Ca 44Ca 46Ca 48Ca
HMulti -13.1 -17.5 -18.1 -13.6 -6.2
HP -9.7 -12.5 -11.9 -8.7 -4.0
HP1 -8.9 -11.8 -11.3 -8.4 -3.9
Hpn
P1 -2.95 -2.2 -1.4 -0.8 -0.2
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
38K
ISOMER SHIFT IN38K
δ r2c (0
+)− δ r2c (3
+) = 0.1fm2
3+GS
0+m
HMulti -7.9 -13.2
HP -4.2 -9.7
HP1 -4.2 -9.2
Hpn
P1 -1.4 -6.1
The strong neutron-proton correlations in 0+
compared to 3+
⇓proton occupancy of fp shell is reduced in 3+
compared to 0+ 7/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
38K
SM RESULTS : ZBM2
✗ bad level scheme ✓ good variation of rc
T=1 matrix elements of the ZBM2 are too strong with respect to theT=0, producing an inversion of the 0+ and 3+
⇓a correction of the ZBM2 is necessary preserving the description ofthe Ca isotopes shift
8/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
38K
SM RESULTS : MODIFIED ZBM2Hm = ∑εi ni +∑aijni ·nj +bij(Ti ·Tj −
3
4nδij)
modified ZBM2 : monopole correction of bij to (d3/2)2 with ∆aij = 0
✓ good level scheme ✓ good variation of rc
The modified version of ZBM2 give us a correct order of the states,with conserving a similar composition of the wave functions.
9/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
PART II : SPECTROSCOPIC PROPERTIES OF THE
NUCLEI AROUND132Sn
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Introduction
Introduction
133Sn 134Sn 135Sn
BE/A keV 8310.091 (18) 8275.160 (24) 8230.687 (23)
Taken from NNDC
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Interest field : Experimental
Experimental interest☞New results of 136,138Sn obtained at RIKEN Nishina center.
136Sn
138Sn
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Interest field : Theoretical
Theoretical interest
G,132Sn core,π(gdsh)⊗ν(hfpi)
SMPN,132Sn core,π(gdsh)⊗ν(hfpi)
Vlow−k , 132Sn core, π(gdsh)⊗ν(hfpi)
13/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Interest field : Astrophysical
Astrophysical interest
☞ responsible of the synthesis of the heavy elements by r-process,and their nuclear model properties predictions give the inputs for
r-process simulations.
100
150
200
250
10
−3
10
−2
10
−110
010
1
Solar r abundances
r−process waiting point (ETFSI−Q)
Known mass
Known half−life
N=184
N=126
N=82
N=50
r−path
28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 5860
6264
666870
727476
7880
8284
86 88 9092 94
9698
100102104
106108
110112114
116118120
122124
126
128130132
134136138
140142
144146
148150
152154
156158
160162
164166
168170
172174
176178
180182184
186188190
26283032
3436
3840
424446
48
50 5254
5658
6062
6466
6870
7274
7678
8082
8486
8890
9294
9698
100
Adapted from K.-L.Kratz
14/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Core and valence space
Core and valence space
15/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Core and valence space
Core and valence space
☞ 1h11/2 and 1g9/2 closed ≡132 Sn core
☞ 1h11/2 and 1g9/2 opened ≡110 Zr core
✓ Opening the 132Sn core constitutes anumerical chalenge in the
diagonalisation of the matrix.
⇓Diagonalization in Antoine ∗ andNathan† codes using Lanczos procedure,
Exemple : 140Sm : D=10 1010
(*)E.Caurier et al, Rev.Mod.Phys 77
(2007)427, and Antoine website
(†)no public version
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
effective interaction
EFFECTIVE INTERACTION
REALISTIC
1. derived from realistic interaction :ArgonneV18, CD-Bonn,N3LO,...
2. renormalised by Vlow−k or G matrix approach to exclude the
repulsive part at short range.
3. adapted to the model space by many body perturbation theory,
usingP̂ and Q̂ projection operators into model space andexcluded space respectively
P = ∑di=1 |Ψi ><Ψi |, Q = ∑∞
i=d |Ψi ><Ψi |, P+Q=1
Veff = V +VQ
E −H0V
︸ ︷︷ ︸
second order
+V QE−H0
V QE−H0
V
V → Vlow−k
M. Hjorth-Jensen et al. Phys.Rep 261 (1995)125
boundary of
lowkV
VNN
0
∞
0∞
Λ
Λ
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
spe
Single particle energies
The re-adjustments of the monopole part of Vlow−k interaction to
obtain the experimental level scheme of 133Sn and 133Sb and theirmasses relative to 132Sn.
NNS110 interaction.18/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
be2
B(E2,6+ → 4+)
✗ NNS110 : predicts well the B(E2)
in 134,138Sn, but it underestimatesit for 136Sn.
0
10
20
30
40
50
133 134 135 136 137 138 139 140 141 142
BE
2(6+
-> 4+)
[e2 .fm4 ]
A
NNS110Exp
eeff (ν) = 0.64e
19/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
be2
B(E2,6+ → 4+)
✗ NNS110 : predicts well the B(E2)
in 134,138Sn, but it underestimatesit for 136Sn.
✗ open core : still underestimated
of 136Sn. 0
10
20
30
40
50
133 134 135 136 137 138 139 140 141 142
BE
2(6+
-> 4+)
[e2 .fm4 ]
A
NNS110Exp
NNS110+(1p1h)
eeff (ν) = 0.5e
eeff (π) = 1.5e
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
be2
B(E2,6+ → 4+)
✗ NNS110 : predicts well the B(E2)in 134,138Sn, but it underestimates
it for 136Sn.
✗ open core : still underestimated
of 136Sn.
✓ reducing the pairing(NNS110P) : gives us a goodagreement with the Exp.
0
10
20
30
40
50
133 134 135 136 137 138 139 140 141 142
BE
2(6+
-> 4+)
[e2 .fm4 ]
A
NNS110Exp
NNS110+(1p1h)NNS110P
H.Naïdja et al . J.Phys.Conf.series, 580 (2015)012030
21/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
seniority
seniority mixing
υ=0 98%
υ=2 95%
υ=2 82%
υ=2 96%
υ=4 84%
0
10
20
30
40
50
133 134 135 136 137 138 139 140 141 142B
E2(
6+->
4+)
[e2 .fm4 ]
A
NNS110Exp
NNS110+(1p1h)
B(E2,υf = υi ,J → J −2) ∝ (1− 2n2j+1 )
2
◮ 6+ and 4+1 are dominately
seniority υ = 2 ⇒ vanishing
B(E2)
◮ 4+2 (υ = 4) is above the 6+
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
seniority
seniority mixing
υ=0 98%
υ=2 93%
υ=2 52%, υ=4 48%
υ=2 45%, υ=4 55%
υ=2 95%
0
10
20
30
40
50
133 134 135 136 137 138 139 140 141 142B
E2(
6+->
4+)
[e2 .fm4 ]
A
NNS110Exp
NNS110+(1p1h)NNS110P
seniority-mixing
B(E2,υf = υi ,J → υi ,J −2) ∝ (1− 2n2j+1 )
2
◮ 4+2 is below the 6+
◮ mixed seniority υ = 2 and υ = 4 in
4+1 and 4+
2 states ⇒ allowed E2transitions.
23/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
seniority
seniority mixing
6+2 (υ = 4) is above the 8+ 6+
2 (υ = 2 and 4) is below the 8+
Pushing down of the 6+2 (υ = 4) state opens up a new channel for the
fast E2 decay of the 8+ state.
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
tins : core
Tins : NNS110 and opening the core
H.Naïdja et al . J.Phys.Conf.series,580 (2015)012030
Small effect of the core excitations (1p1h) to the level scheme of tin isotopes.
The robust nature of 132Sn core 25/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
tins :NNS
Tins : NNS110P interaction
ν(f7/2)2 ≈80%-96% ν(f7/2)
4 ≈60%-80% ν(f7/2)6 ≈50%-70%
H.Naïdja et al . J.Phys.Conf.series, 580 (2015)012030
Reducing the pairing strength (NNS110P interaction), improves
clearly the agreement with the data. 26/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
masses
Masses◮ Binding energies relative to 132Sn
-30
-25
-20
-15
-10
-5
0
5
132 133 134 135 136 137 138 139 140
BE
rel
ativ
e to
13
2 Sn
[MeV
]
A
ExpNNS110P
NNS110Vlowk
◮ one neutron separation energy
0
500
1000
1500
2000
2500
3000
3500
4000
4500
133 134 135 136 137S
n[ke
V]
A
ExpNNS110
NNS110PVlowk
H.Naïdja et al . J.Phys.Conf.series, 580 (2015)012030
✓ Our masses are consistent with the data
✓ The 2 body monopole corrections are believed to come from 3
body interaction not included in Vlow−k27/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
sub-shell closure at N=90 ?
28/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
bref
sub-shell closure at N=90 ?
Taken
from
O.Sorlin
notes
:
◮ Analogy between 22O, 48Ca, and 140Sn in theclosure of the (sub-)shell at N=14,28, and 90 ?
A.P.Zuker, PRL 90, 042502 (2003)T.Otsuka et al. PRL 105,032501 (2010) 29/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Sn140
Sub-shell closure at N=90 ?ν(f7/2)
6(p3/2)2
ν(f7/2)7(p3/2)
1
ν(f7/2)6(p3/2)
2
ν(f7/2)8
0
10
20
30
40
50
133 134 135 136 137 138 139 140 141 142B
E2(
6+->
4+)
[e2 .fm4 ]
A
NNS110Exp
NNS110P
H.Naïdja and al . J.Phys.Conf series, 580 (2015)012030
◮ the excited states are characterized by mixed configurations.
our predictions to 140Sn structure using NNS110P
30/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Sn140
Sub-shell closure at N=90 ?
0
0.5
1
1.5
2
2.5
3
133 134 135 136 137 138 139 140 141
E [M
eV]
N
2+
4+
6+
8+
H.Naïdja et al . J.Phys.Conf.series, 580 (2015)012030
◮ the spacing 0+−2+ remainsnearly constant at around 700
keV, except for a small increase at140Sn owing to the filling of the(f7/2).
S.Sarkar and M.S.Sarkar Phys.Rev.C 81, 064328(2010)
◮ a sudden increase for N=90,
indicating a closed-shell structurefor 140Sn.
2+ energy in 140Sn in not higher than in the neighboring nuclei. 31/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
ESPE
Evolution of neutron ESPE
-0.5
0
0.5
1
1.5
133 134 135 136 137 138 139 140 141
espe
(MeV
)
A
f7/2
p3/2
H.Naïdja et al . J.Phys.Conf.series, 580 (2015)012030
◮ The gap between νf7/2 and νp3/2
remains constant
⇓No sub-shell closure at N=90
S.Sarkar and M.S.Sarkar Phys.Rev.C 81, 064328(2010)
◮ The gap νf7/2 and νp3/2
increases to 2.246 MeV⇓
140Sn is doubly magic nucleus
2.2 MeV880 keV
750 keV
32/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
ESPE
Evolution of neutron ESPE
-1
-0.5
0
0.5
1
1.5
2
133 134 135 136 137 138 139 140 141
espe
(MeV
)
A
f7/2
p3/2
◮ Increasing the gap by changing
the monopole part
S.Sarkar and M.S.Sarkar Phys.Rev.C 81, 064328(2010)
◮ The gap νf7/2 and νp3/2
increases to 2.246 MeV
⇓140Sn is a doubly magic nucleus
2.2 MeV2 MeV
33/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
gap
Increasing the gap
0
10
20
30
40
50
133 134 135 136 137 138 139 140 141 142
BE
2(6+
-> 4+)
[e2 .fm4 ]
A
NNS110PExp
increasing gap
Increasing the gap by changing the monopole part
⇓◮ Transitions are slightly inconsistent with the experience◮ sudden decrease of B(E2) in 140Sn
1. E(2) favor the transitions from j → j +2(p3/2 → f7/2) compared as
j → j +1(f5/2 → f7/2)
ν(f7/2)7(f5/2)
1
ν(f7/2)7(p3/2)
1
ν(f7/2)7(p3/2)
1
ν(f7/2)8
34/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
gap
Tins : increasing the gap
increasing f7/2 −p3/2 gap, has an apparent effect to 138Sn structure.
⇓Losing the spectroscopic properties.
No sub-shell closure at N=9035/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
gap
OTHER APPLICATIONS TO THE OPEN N-P SYSTEMS
36/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Te
Tellerium
37/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
xe
Xenon
38/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Ba
Barium
39/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Ce
Cerium
40/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Nd
Neodymium
41/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Be
Collective propertiesB(E2,2+ → 0+)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
134Te136
Te138
Te136
Xe138
Xe140
Xe138
Ba140
Ba142
Ba140
Ce142
Ce144
Ce142
Nd144
Nd146
Nd
BE
2(2+
-> 0+)
[e2 .fm4 ]
A
NNS110PExp
✓ B(E2) are reporduced with no explicit adjustement of the eeff .
✓ BE2 in 136Te is close to that 134Te, so anomaly in 136Te
eeff (ν) = 0.5e
eeff (π) = 1.5e
Tobe
published
42/44
outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
Triaxiality
Triaxiality◮ Q(2+
γ ) =−Q(2+yrast ).
◮ the presence of B(E2,2+γ → 2+
1 ) transition
◮ Q(3+) = 0.
◮ strong B(E2,3+ → 2+2 ) transition
work under progress with Bounseng Bounthong using the deformed
HF with constraints
138Te 140Xe 142Ba 144Ce 146Nd
Q(2+1)e.fm2 -45.67 -62.64 -70.74 -75.24 -61.76
Q(2+2 )e.fm2 40.08 63.84 69.18 75.49 -60.92
Q(3+)e.fm2 -0.34 -1.31 -0.62 -0.79 -0.10
B(2+2 → 2+
1 )e2.fm4 43 155 180 210 294
B(3+ → 2+2)e2.fm4 745 1911 2054 2569 2153
Qi(Q0) 157 220 248 262 248Qi(B(E2)) 160 225 252 268 255β 0.1 0.14 0.15 0.15 0.14
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outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions
CONCLUSIONS-PERSPECTIVES
✓ Using NNS110P interaction, the agreement between the experience andcalculated energy levels is improved.
✓ The pairing force must be reduced to reproduce the experimental transition ratesin 136Sn, leading to mixing seniority.
✓ The core excitations seem to have a negligible effect to the tin isotopes energies,confirming the strong magicity of 132Sn
✓ 140Sn doesn’t exhibit the features of a doubly magic nucleus.
✓ The applications to other nuclei allowed us to test our interaction to differentssystems.
PROJECTS UNDER PROGRESS
◮ Triaxiality in 140Sm : in collaboration with Andreas Görgen, University of Oslo.
◮ Isomer in 140Sb : in collaboration with Radomira Lozeva ; IPHC Strasbourg
◮ High spin states in 138Te and 140Xe : in collaboration with W.Urban,
◮ The deformation in the nuclei beyond 132Sn with B.Bounthong.
◮ ...
44/44